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%I #50 Jan 31 2019 08:06:11
%S 0,1,2,3,5,4,8,12,20,8,16,24,40,32,64,88,152,16,32,48,80,64,128,192,
%T 320,128,256,384,640,512,1024,1360,2384,32,64,96,160,128,256,384,640,
%U 256,512,768,1280,1024,2048,2816,4864,512,1024,1536,2560,2048,4096,6144,10240,4096
%N Number of chiral partitions of n; number of irreducible representations of the symmetric group S_n with nontrivial determinant.
%H Amritanshu Prasad, <a href="/A272090/b272090.txt">Table of n, a(n) for n = 1..9999</a>
%H Arvind Ayyer, Amritanshu Prasad, Steven Spallone, <a href="https://arxiv.org/abs/1604.08837">Representations of symmetric groups with non-trivial determinant</a>, arXiv:1604.08837 [math.RT], 2016.
%H Arvind Ayyer, Amritanshu Prasad, Steven Spallone, <a href="https://arxiv.org/abs/1812.00608">Macdonald trees and determinants of representations for finite Coxeter groups</a>, arXiv:1812.00608 [math.RT], 2018.
%H Amritanshu Prasad, <a href="/A272090/a272090.sage.txt">Sage program</a>
%F a(n) = A000041(n) - A045923(n).
%F If n = e + Sum_{i=1..r}2^ki in binary expansion, with e=0 or 1, 0<k1<...<kr, then a(n) = 2^(Sum_{i=2..r}k_i)(2^(k1-1)+Sum_{v=1..k1-1}2^((v+1)(k1-2)-v(v-1)/2)+e2^(k1(k1-1)/2).
%e The sign representation and the two-dimensional representation of S_3 have nontrivial determinant, so a(3)=2.
%t a[1] = 0;
%t a[n_] := Module[{bb, e, pos, k, r}, bb = Reverse[IntegerDigits[n, 2]]; e = bb[[1]]; pos = DeleteCases[Flatten[Position[bb, 1]], 1]-1; r = Length[ pos]; Do[k[i] = pos[[i]], {i, r}]; 2^Sum[k[i], {i, 2, r}] (2^(k[1]-1) + Sum[2^((v+1)(k[1]-2)-v(v-1)/2), {v, k[1]-1}] + e 2^(k[1] (k[1]-1)/2))];
%t Array[a, 60] (* _Jean-François Alcover_, Aug 09 2018 *)
%o (PARI) a(n) = {if (n==1, 0, if (n % 2, ns = n-1; eps = 1, ns = n; eps = 0); b = Vecrev(binary(ns/2)); vk = select(x->(x != 0), b, 1); k1 = vk[1]; 2^sum(i=2, #vk, vk[i])*(2^(k1-1) + sum(v=1, k1-1, 2^((v+1)*(k1-2)-binomial(v,2))) + eps*2^binomial(k1,2)););} \\ _Michel Marcus_, May 11 2016
%Y Cf. A000041, A045923.
%K nonn
%O 1,3
%A _Amritanshu Prasad_, May 10 2016
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